Slippery Slides - UKR · A toy example General advice for slides Clara L oh 2. An axiom for slides...

Slippery Slides

Clara Loh

Universitat Regensburg

January 21, 2018

Overview

An axiom for slides

A toy example

General advice for slides

Clara Loh 2

An axiom for slides

Axiom

Do not give slide talks!

〈audience growls〉

Except if . . .

I there is no blackboard available

I the material cannot be displayed on a blackboard(e.g., animations, . . . )

I you know exactly what you are doing.

Question

What is so difficult about giving good slide talks?!

Clara Loh An axiom for slides 3

An axiom for slides

Axiom

Do not give slide talks!

〈audience growls〉

Except if . . .

I there is no blackboard available

I the material cannot be displayed on a blackboard(e.g., animations, . . . )

I you know exactly what you are doing.

Question

What is so difficult about giving good slide talks?!

Clara Loh An axiom for slides 3

An axiom for slides

Axiom

Do not give slide talks!

〈audience growls〉

Except if . . .

I there is no blackboard available

I the material cannot be displayed on a blackboard(e.g., animations, . . . )

I you know exactly what you are doing.

Question

What is so difficult about giving good slide talks?!

Clara Loh An axiom for slides 3

An axiom for slides

Axiom

Do not give slide talks!

〈audience growls〉

Except if . . .

I there is no blackboard available

I the material cannot be displayed on a blackboard(e.g., animations, . . . )

I you know exactly what you are doing.

Question

What is so difficult about giving good slide talks?!

Clara Loh An axiom for slides 3

An axiom for slides

Axiom

Do not give slide talks!

〈audience growls〉

Except if . . .

I there is no blackboard available

I the material cannot be displayed on a blackboard(e.g., animations, . . . )

I you know exactly what you are doing.

Question

What is so difficult about giving good slide talks?!

Clara Loh An axiom for slides 3

An axiom for slides

Axiom

Do not give slide talks!

〈audience growls〉

Except if . . .

I there is no blackboard available

I the material cannot be displayed on a blackboard(e.g., animations, . . . )

I you know exactly what you are doing.

Question

What is so difficult about giving good slide talks?!

Clara Loh An axiom for slides 3

An axiom for slides

Axiom

Do not give slide talks!

〈audience growls〉

Except if . . .

I there is no blackboard available

I the material cannot be displayed on a blackboard(e.g., animations, . . . )

I you know exactly what you are doing.

Question

What is so difficult about giving good slide talks?!

Clara Loh An axiom for slides 3

What is so difficult about slides?

Problem

The speaker needs toforesee/manipulate the future.

I fixed content, length, and order of talk (up to minor variations)

I difficult to insert comments/correct typos

I hence: limited interaction with the audience

Solution

I Meticulous planning

I Leading/manipulating the audience (requires experience)

Clara Loh An axiom for slides 4

What is so difficult about slides?

Problem

The speaker needs toforesee/manipulate the future.

I fixed content, length, and order of talk (up to minor variations)

I difficult to insert comments/correct typos

I hence: limited interaction with the audience

Solution

I Meticulous planning

I Leading/manipulating the audience (requires experience)

Clara Loh An axiom for slides 4

What is so difficult about slides?

Problem

The speaker needs toforesee/manipulate the future.

I fixed content, length, and order of talk (up to minor variations)

I difficult to insert comments/correct typos

I hence: limited interaction with the audience

Solution

I Meticulous planning

I Leading/manipulating the audience (requires experience)

Clara Loh An axiom for slides 4

What is so difficult about slides?

Problem

The speaker needs toforesee/manipulate the future.

I fixed content, length, and order of talk (up to minor variations)

I difficult to insert comments/correct typos

I hence: limited interaction with the audience

Solution

I Meticulous planning

I Leading/manipulating the audience (requires experience)

Clara Loh An axiom for slides 4

What is so difficult about slides?

Problem

The speaker needs toforesee/manipulate the future.

I fixed content, length, and order of talk (up to minor variations)

I difficult to insert comments/correct typos

I hence: limited interaction with the audience

Solution

I Meticulous planning

I Leading/manipulating the audience (requires experience)

Clara Loh An axiom for slides 4

What is so difficult about slides?

Problem

I The human brain has limitedprocessing speed/capacity.

I By using slides, the audiencedoes not magically getsmarter/faster!

Bad reason for a slide talk

“I only have x minutes for the talk.

On a blackboard, I would get nowhere. But, using slides, . . . ”

Solution

Careful selection/presentation of topics

Clara Loh An axiom for slides 5

What is so difficult about slides?

Problem

I The human brain has limitedprocessing speed/capacity.

I By using slides, the audiencedoes not magically getsmarter/faster!

Bad reason for a slide talk

“I only have x minutes for the talk.On a blackboard, I would get nowhere. But, using slides, . . . ”

Solution

Careful selection/presentation of topics

Clara Loh An axiom for slides 5

What is so diflicuft about slides?

Problem

I The human brain has limitedprocessing speed/capacity.

I By using slides, the audiencedoes not magically getsmarter/faster!

Bad reason for a slide talk

“I only have x minutes for the talk.On a blackboard, I would get nowhere. But, using slides, . . . ”

Solution

Careful selection/presentation of topics

Clara Loh An axiom for slides 5

What is so difficult about slides?

Problem

I The human brain has limitedprocessing speed/capacity.

I By using slides, the audiencedoes not magically getsmarter/faster!

Bad reason for a slide talk

“I only have x minutes for the talk.On a blackboard, I would get nowhere. But, using slides, . . . ”

Solution

Careful selection/presentation of topics

Clara Loh An axiom for slides 5

A little experiment

On the previous slide, there was

a witch a brain a funnel a hose

Question

I How many hands/feet did the witch have?

I How many knots did the hose have?

I Did the title of the previous slide change?

Conclusion

The audience needs to be told what to look at!

Clara Loh An axiom for slides 6

A little experiment

On the previous slide, there was

a witch a brain a funnel a hose

Question

I How many hands/feet did the witch have?

I How many knots did the hose have?

I Did the title of the previous slide change?

Conclusion

The audience needs to be told what to look at!

Clara Loh An axiom for slides 6

A little experiment

On the previous slide, there was

a witch a brain a funnel a hose

Question

I How many hands/feet did the witch have?

I How many knots did the hose have?

I Did the title of the previous slide change?

Conclusion

The audience needs to be told what to look at!Clara Loh An axiom for slides 6

A toy example

Let’s look at an example . . .

Clara Loh A toy example 7

Solving the Blorxification Equation

Nimeta Kulmkapp

University of Dokomo

21.01.2018

The Blorxification Equation

The Blorxification Equation is the equation for a tectonic phenomenonon the planet Blorx. It is noteworthy that planet Blorx has the shapeof a cube. A version with slightly different constants has previouslybeen studied in a paper by C. Keen (2017, unpublished).

Blorxification Equation:√2 + x + cos(8 · y) + z2018 = x · 2018

x + y + z = 3 · yz = x

−1 ≤ x , y ≤ 1

−1 ≤ z ≤ 1

Theorem

The Blorxification Equation has a solution.!

The value of your SchummerCoin wallet

has dropped by 250% !

!New Message

from DubiousMadame !

N. Kulmkapp 2 / 561

The Blorxification Equation

The Blorxification Equation is the equation for a tectonic phenomenonon the planet Blorx. It is noteworthy that planet Blorx has the shapeof a cube. A version with slightly different constants has previouslybeen studied in a paper by C. Keen (2017, unpublished).Blorxification Equation:√

2 + x + cos(8 · y) + z2018 = x · 2018

x + y + z = 3 · yz = x

−1 ≤ x , y ≤ 1

−1 ≤ z ≤ 1

Theorem

The Blorxification Equation has a solution.!

The value of your SchummerCoin wallet

has dropped by 250% !

!New Message

from DubiousMadame !

N. Kulmkapp 2 / 561

The Blorxification Equation

The Blorxification Equation is the equation for a tectonic phenomenonon the planet Blorx. It is noteworthy that planet Blorx has the shapeof a cube. A version with slightly different constants has previouslybeen studied in a paper by C. Keen (2017, unpublished).Blorxification Equation:√

2 + x + cos(8 · y) + z2018 = x · 2018

x + y + z = 3 · yz = x

−1 ≤ x , y ≤ 1

−1 ≤ z ≤ 1

Theorem

The Blorxification Equation has a solution.!

The value of your SchummerCoin wallet

has dropped by 250% !

!New Message

from DubiousMadame !

N. Kulmkapp 2 / 561

Proofs

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 3 / 561

More proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.

N. Kulmkapp 4 / 561

Even more proofs

Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

N. Kulmkapp 5 / 561

Jumping proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 6 / 561

Jumping proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 6 / 561

Jumping proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 6 / 561

Jumping proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 6 / 561

Jumping proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 6 / 561

Jumping proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 6 / 561

Jumping proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 6 / 561

Jumping proofs

Small text successfully keeps the audience from reading it.This is even worse. Using this fontsize, we can pack a quasi-infinite amount of information on a single slide.Moreover, this should be almost as bad as smallprint in mobile phone contracts. In combination with a lowprojector resolution, the result can be quite impressive.

We will skip these slides. They are not so important.But, if they are not important, why are they in the slide deck?Because we skip these slides anyway, we can also use yellow forhighlighting.Of course, we can also happily skip some computations:

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 − 1 dx = 0,

∫ 1

01 + 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1 + 1 dx = 2.

Theorem

These slides will not be shown.

N. Kulmkapp 6 / 561

Proof of the main result

The function

f (a, b, c) =

1/2018 ·√a + c2018 + 2 + cos(8 · b)

1/4 · (a + b + c)a

satisfies the hypotheses of the Blorx Fixed Point Theorem. This solvesthe Blorxification equation.

N. Kulmkapp 7 / 561

A toy example

Let’s try to do better . . .

Clara Loh A toy example 8

Solving the Blorxification Equation

Nimeta Kulmkapp

University of Dokomo

21. 01. 2018

Joint project with Jargmine Peatus

Blorxification

Tectonic phenomenon on the cube-shaped planet Blorx

modelled by the Blorx operator

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Blorxcillationaveragerandom swap

Question

Are there points on Blorx that are not affected by Blorxification?

Theorem (K., J. Peatus 2018)

The Blorx operator B : Q −→ Q has a fixed point.

Related Work

I C. Keen treated a similar operator (2017)

I our proof uses the same method

Clara Loh A toy example 10

Blorxification

Tectonic phenomenon on the cube-shaped planet Blorxmodelled by the Blorx operator

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Blorxcillationaveragerandom swap

Question

Are there points on Blorx that are not affected by Blorxification?

Theorem (K., J. Peatus 2018)

The Blorx operator B : Q −→ Q has a fixed point.

Related Work

I C. Keen treated a similar operator (2017)

I our proof uses the same method

Clara Loh A toy example 10

Blorxification

Tectonic phenomenon on the cube-shaped planet Blorxmodelled by the Blorx operator

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Blorxcillation

averagerandom swap

Question

Are there points on Blorx that are not affected by Blorxification?

Theorem (K., J. Peatus 2018)

The Blorx operator B : Q −→ Q has a fixed point.

Related Work

I C. Keen treated a similar operator (2017)

I our proof uses the same method

Clara Loh A toy example 10

Blorxification

Tectonic phenomenon on the cube-shaped planet Blorxmodelled by the Blorx operator

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Blorxcillationaverage

random swap

Question

Are there points on Blorx that are not affected by Blorxification?

Theorem (K., J. Peatus 2018)

The Blorx operator B : Q −→ Q has a fixed point.

Related Work

I C. Keen treated a similar operator (2017)

I our proof uses the same method

Clara Loh A toy example 10

Blorxification

Tectonic phenomenon on the cube-shaped planet Blorxmodelled by the Blorx operator

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Blorxcillationaveragerandom swap

Question

Are there points on Blorx that are not affected by Blorxification?

Theorem (K., J. Peatus 2018)

The Blorx operator B : Q −→ Q has a fixed point.

Related Work

I C. Keen treated a similar operator (2017)

I our proof uses the same method

Clara Loh A toy example 10

Blorxification

Tectonic phenomenon on the cube-shaped planet Blorxmodelled by the Blorx operator

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Blorxcillationaveragerandom swap

Question

Are there points on Blorx that are not affected by Blorxification?

Theorem (K., J. Peatus 2018)

The Blorx operator B : Q −→ Q has a fixed point.

Related Work

I C. Keen treated a similar operator (2017)

I our proof uses the same method

Clara Loh A toy example 10

Blorxification

Tectonic phenomenon on the cube-shaped planet Blorxmodelled by the Blorx operator

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Blorxcillationaveragerandom swap

Question

Are there points on Blorx that are not affected by Blorxification?

Theorem (K., J. Peatus 2018)

The Blorx operator B : Q −→ Q has a fixed point.

Related Work

I C. Keen treated a similar operator (2017)

I our proof uses the same method

Clara Loh A toy example 10

Blorxification

Tectonic phenomenon on the cube-shaped planet Blorxmodelled by the Blorx operator

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Blorxcillationaveragerandom swap

Question

Are there points on Blorx that are not affected by Blorxification?

Theorem (K., J. Peatus 2018)

The Blorx operator B : Q −→ Q has a fixed point.

Related Work

I C. Keen treated a similar operator (2017)

I our proof uses the same method

Clara Loh A toy example 10

Proof of the main result

The Blorx operator B has a fixed point, where

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Proof.

I Idea: Use the Blorx Fixed Point Theorem:Every continuous map Q −→ Q has a fixed point.

I Check that the map B : Q −→ Q is continuous.

I Thus, by the Blorx Fixed Point Theorem, B has a fixed point.

Clara Loh A toy example 11

Proof of the main result

The Blorx operator B has a fixed point, where

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Proof.

I Idea: Use the Blorx Fixed Point Theorem:Every continuous map Q −→ Q has a fixed point.

I Check that the map B : Q −→ Q is continuous.

I Thus, by the Blorx Fixed Point Theorem, B has a fixed point.

Clara Loh A toy example 11

Proof of the main result

The Blorx operator B has a fixed point, where

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Proof.

I Idea: Use the Blorx Fixed Point Theorem:Every continuous map Q −→ Q has a fixed point.

I Check that the map B : Q −→ Q is continuous.

I Thus, by the Blorx Fixed Point Theorem, B has a fixed point.

Clara Loh A toy example 11

Proof of the main result

The Blorx operator B has a fixed point, where

B : [−1, 1]3 −→ [−1, 1]3 =: Qxyz

7−→

12018 ·

√2 + x + cos(8 · y) + z201813 · (x + y + z)

x

Proof.

I Idea: Use the Blorx Fixed Point Theorem:Every continuous map Q −→ Q has a fixed point.

I Check that the map B : Q −→ Q is continuous.

I Thus, by the Blorx Fixed Point Theorem, B has a fixed point.

Clara Loh A toy example 11

Excursion: Categories

Definition (Category: models relations between objects)

A category C consists of the following data

I set class Ob(C ): the objects of C

I sets MorC (X ,Y ): the morphisms X −→ Y

I compositions ◦ : MorC (Y ,Z )×MorC (X ,Y ) −→ MorC (X ,Z )

satisfying the following conditions:

I ◦ is associative

I existence of left/right neutral morphisms idX ∈ MorC (X ,X )

X

idXf

Y

g

Z

g ◦ f

Clara Loh A toy example 12

Excursion: Categories

Definition (Category: models relations between objects)

A category C consists of the following data

I set class Ob(C ): the objects of C

I sets MorC (X ,Y ): the morphisms X −→ Y

I compositions ◦ : MorC (Y ,Z )×MorC (X ,Y ) −→ MorC (X ,Z )

satisfying the following conditions:

I ◦ is associative

I existence of left/right neutral morphisms idX ∈ MorC (X ,X )

X

idX

fY

gZ

g ◦ f

Clara Loh A toy example 12

Excursion: Categories

Definition (Category: models relations between objects)

A category C consists of the following data

I set class Ob(C ): the objects of C

I sets MorC (X ,Y ): the morphisms X −→ Y

I compositions ◦ : MorC (Y ,Z )×MorC (X ,Y ) −→ MorC (X ,Z )

satisfying the following conditions:

I ◦ is associative

I existence of left/right neutral morphisms idX ∈ MorC (X ,X )

X

idX

fY

gZ

g ◦ f

Clara Loh A toy example 12

Excursion: Categories

Definition (Category: models relations between objects)

A category C consists of the following data

I set class Ob(C ): the objects of C

I sets MorC (X ,Y ): the morphisms X −→ Y

I compositions ◦ : MorC (Y ,Z )×MorC (X ,Y ) −→ MorC (X ,Z )

satisfying the following conditions:

I ◦ is associative

I existence of left/right neutral morphisms idX ∈ MorC (X ,X )

X

idX

fY

gZ

g ◦ f

Clara Loh A toy example 12

Excursion: Categories

Definition (Category: models relations between objects)

A category C consists of the following data

I set class Ob(C ): the objects of C

I sets MorC (X ,Y ): the morphisms X −→ Y

I compositions ◦ : MorC (Y ,Z )×MorC (X ,Y ) −→ MorC (X ,Z )

satisfying the following conditions:

I ◦ is associative

I existence of left/right neutral morphisms idX ∈ MorC (X ,X )

X

idXf

Yg

Z

g ◦ f

Clara Loh A toy example 12

Excursion: Categories

Definition (Category: models relations between objects)

A category C consists of the following data

I set class Ob(C ): the objects of C

I sets MorC (X ,Y ): the morphisms X −→ Y

I compositions ◦ : MorC (Y ,Z )×MorC (X ,Y ) −→ MorC (X ,Z )

satisfying the following conditions:

I ◦ is associative

I existence of left/right neutral morphisms idX ∈ MorC (X ,X )

Example

Linear Algebra VectR R-vector spaces R-linear mapsTopology Top topological spaces continuous maps

Clara Loh A toy example 12

Excursion: Functors

Definition (Functor: translates between categories)

Let C , D be categories.A contravariant functor F : C −→ D consists of the following data

I map F : Ob(C ) −→ Ob(D)

I maps F : MorC (X ,Y ) −→ MorC (F (Y ),F (X ))

satisfying the following conditions:

I F (idX ) = idF (X )

I F (g ◦ f ) = F (f ) ◦ F (g)

X Y

f

F (X ) F (Y )

F (f )

Clara Loh A toy example 13

Excursion: Functors

Definition (Functor: translates between categories)

Let C , D be categories.A contravariant functor F : C −→ D consists of the following data

I map F : Ob(C ) −→ Ob(D)

I maps F : MorC (X ,Y ) −→ MorC (F (Y ),F (X ))

satisfying the following conditions:

I F (idX ) = idF (X )

I F (g ◦ f ) = F (f ) ◦ F (g)

X Yf

F (X ) F (Y )F (f )

Clara Loh A toy example 13

Excursion: Functors

Definition (Functor: translates between categories)

Let C , D be categories.A contravariant functor F : C −→ D consists of the following data

I map F : Ob(C ) −→ Ob(D)

I maps F : MorC (X ,Y ) −→ MorC (F (Y ),F (X ))

satisfying the following conditions:

I F (idX ) = idF (X )

I F (g ◦ f ) = F (f ) ◦ F (g)

X Yf

F (X ) F (Y )F (f )

Clara Loh A toy example 13

Excursion: Functors

Definition (Functor: translates between categories)

Let C , D be categories.A contravariant functor F : C −→ D consists of the following data

I map F : Ob(C ) −→ Ob(D)

I maps F : MorC (X ,Y ) −→ MorC (F (Y ),F (X ))

satisfying the following conditions:

I F (idX ) = idF (X )

I F (g ◦ f ) = F (f ) ◦ F (g)

X Yf

F (X ) F (Y )F (f )

Clara Loh A toy example 13

Excursion: Functors

Definition (Functor: translates between categories)

Let C , D be categories.A contravariant functor F : C −→ D consists of the following data

I map F : Ob(C ) −→ Ob(D)

I maps F : MorC (X ,Y ) −→ MorC (F (Y ),F (X ))

satisfying the following conditions:

I F (idX ) = idF (X )

I F (g ◦ f ) = F (f ) ◦ F (g)

Example

I dual vector space HomR( · ,R) : VectR −→ VectR

I singular cohomology H2 : Top −→ VectR such that

H2(Q) ∼= 0 and H2(∂Q) ∼= R.

Clara Loh A toy example 13

Excursion: Functors

Definition (Functor: translates between categories)

Let C , D be categories.A contravariant functor F : C −→ D consists of the following data

I map F : Ob(C ) −→ Ob(D)

I maps F : MorC (X ,Y ) −→ MorC (F (Y ),F (X ))

satisfying the following conditions:

I F (idX ) = idF (X )

I F (g ◦ f ) = F (f ) ◦ F (g)

Example

I dual vector space HomR( · ,R) : VectR −→ VectRI singular cohomology H2 : Top −→ VectR such that

H2(Q) ∼= 0 and H2(∂Q) ∼= R.Clara Loh A toy example 13

Proof of the Blorx Fixed Point Theorem

Every continuous map Q −→ Q has a fixed point.

Proof by contradiction.

I Assume: there is a continuous f : Q −→ Q without fixed point.

I Use f to construct a continuous map g : Q −→ ∂Q

with g ◦ (inclusion ∂Q ↪→ Q) = id∂Q .

f (x)

xg(x)

I Apply the functor H2 : Top −→ VectR:

∂Qid∂Q//

��

∂Q

Qg

==

R ∼=

H2(∂Q) H2(∂Q)

∼= R

H2(id∂Q)

=id

oo

H2(g)uu

0 ∼=

H2(Q)

OO

I Hence, idR factors over 0. Contradiction!

Clara Loh A toy example 14

Proof of the Blorx Fixed Point Theorem

Every continuous map Q −→ Q has a fixed point.

Proof by contradiction.

I Assume: there is a continuous f : Q −→ Q without fixed point.

I Use f to construct a continuous map g : Q −→ ∂Q

with g ◦ (inclusion ∂Q ↪→ Q) = id∂Q .

f (x)

x

g(x)

I Apply the functor H2 : Top −→ VectR:

∂Qid∂Q//

��

∂Q

Qg

==

R ∼=

H2(∂Q) H2(∂Q)

∼= R

H2(id∂Q)

=id

oo

H2(g)uu

0 ∼=

H2(Q)

OO

I Hence, idR factors over 0. Contradiction!

Clara Loh A toy example 14

Proof of the Blorx Fixed Point Theorem

Every continuous map Q −→ Q has a fixed point.

Proof by contradiction.

I Assume: there is a continuous f : Q −→ Q without fixed point.

I Use f to construct a continuous map g : Q −→ ∂Q

with g ◦ (inclusion ∂Q ↪→ Q) = id∂Q .

f (x)

x

g(x)

I Apply the functor H2 : Top −→ VectR:

∂Qid∂Q//

��

∂Q

Qg

==

R ∼=

H2(∂Q) H2(∂Q)

∼= R

H2(id∂Q)

=id

oo

H2(g)uu

0 ∼=

H2(Q)

OO

I Hence, idR factors over 0. Contradiction!

Clara Loh A toy example 14

Proof of the Blorx Fixed Point Theorem

Every continuous map Q −→ Q has a fixed point.

Proof by contradiction.

I Assume: there is a continuous f : Q −→ Q without fixed point.

I Use f to construct a continuous map g : Q −→ ∂Qwith g ◦ (inclusion ∂Q ↪→ Q) = id∂Q .

f (x)

xg(x)

I Apply the functor H2 : Top −→ VectR:

∂Qid∂Q//

��

∂Q

Qg

==

R ∼=

H2(∂Q) H2(∂Q)

∼= R

H2(id∂Q)

=id

oo

H2(g)uu

0 ∼=

H2(Q)

OO

I Hence, idR factors over 0. Contradiction!

Clara Loh A toy example 14

Proof of the Blorx Fixed Point Theorem

Every continuous map Q −→ Q has a fixed point.

Proof by contradiction.

I Assume: there is a continuous f : Q −→ Q without fixed point.

I Use f to construct a continuous map g : Q −→ ∂Qwith g ◦ (inclusion ∂Q ↪→ Q) = id∂Q .

f (x)

xg(x)

I Apply the functor H2 : Top −→ VectR:

∂Qid∂Q//

��

∂Q

Qg

==

R ∼=

H2(∂Q) H2(∂Q)

∼= R

H2(id∂Q)

=id

oo

H2(g)uu

0 ∼=

H2(Q)

OO

I Hence, idR factors over 0. Contradiction!

Clara Loh A toy example 14

Proof of the Blorx Fixed Point Theorem

Every continuous map Q −→ Q has a fixed point.

Proof by contradiction.

I Assume: there is a continuous f : Q −→ Q without fixed point.

I Use f to construct a continuous map g : Q −→ ∂Qwith g ◦ (inclusion ∂Q ↪→ Q) = id∂Q .

f (x)

xg(x)

I Apply the functor H2 : Top −→ VectR:

∂Qid∂Q//

��

∂Q

Qg

==

R ∼=

H2(∂Q) H2(∂Q)

∼= R

H2(id∂Q)

=id

oo

H2(g)uu

0 ∼=

H2(Q)

OO

I Hence, idR factors over 0. Contradiction!

Clara Loh A toy example 14

Proof of the Blorx Fixed Point Theorem

Every continuous map Q −→ Q has a fixed point.

Proof by contradiction.

I Assume: there is a continuous f : Q −→ Q without fixed point.

I Use f to construct a continuous map g : Q −→ ∂Qwith g ◦ (inclusion ∂Q ↪→ Q) = id∂Q .

f (x)

xg(x)

I Apply the functor H2 : Top −→ VectR:

∂Qid∂Q//

��

∂Q

Qg

==R ∼= H2(∂Q) H2(∂Q) ∼= R

H2(id∂Q)=idoo

H2(g)uu

0 ∼= H2(Q)

OO

I Hence, idR factors over 0. Contradiction!

Clara Loh A toy example 14

Proof of the Blorx Fixed Point Theorem

Every continuous map Q −→ Q has a fixed point.

Proof by contradiction.

I Assume: there is a continuous f : Q −→ Q without fixed point.

I Use f to construct a continuous map g : Q −→ ∂Qwith g ◦ (inclusion ∂Q ↪→ Q) = id∂Q .

f (x)

xg(x)

I Apply the functor H2 : Top −→ VectR:

∂Qid∂Q//

��

∂Q

Qg

==R ∼= H2(∂Q) H2(∂Q) ∼= R

H2(id∂Q)=idoo

H2(g)uu

0 ∼= H2(Q)

OO

I Hence, idR factors over 0. Contradiction!

Clara Loh A toy example 14

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides

(the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Slide etiquette

I Use sans serif fonts, matching math fonts

I No fancy firlifanz

I Use high-resolution/vector graphics, explain graphics

I Avoid long sentences

I Break long formulae into small building blocks

I At most 12 lines per slide (for appropriate choice of 12)

I Be careful with the last line on each slide (duration of visibility!)

I Repeat important facts from previous slides

I Do not jump many slides at once“This is not important right now.” “We don’t have time for this.”

I Maximal number of slides (the larger 2 is, the better):

# minutes

2

Clara Loh General advice for slides 15

Technical etiquette

I Bring the correct plugs to connect computer and projector

I Connect computer and projector ahead of time

I Know how to switch to fullscreen/presentation mode

I Switch off system notifications

I Switch off networking (unless necessary for the presentation)

I Do not point to the screen of the computer,but to the projected image

Clara Loh General advice for slides 16

Technical etiquette

I Bring the correct plugs to connect computer and projector

I Connect computer and projector ahead of time

I Know how to switch to fullscreen/presentation mode

I Switch off system notifications

I Switch off networking (unless necessary for the presentation)

I Do not point to the screen of the computer,but to the projected image

Clara Loh General advice for slides 16

Technical etiquette

I Bring the correct plugs to connect computer and projector

I Connect computer and projector ahead of time

I Know how to switch to fullscreen/presentation mode

I Switch off system notifications

I Switch off networking (unless necessary for the presentation)

I Do not point to the screen of the computer,but to the projected image

Clara Loh General advice for slides 16

Technical etiquette

I Bring the correct plugs to connect computer and projector

I Connect computer and projector ahead of time

I Know how to switch to fullscreen/presentation mode

I Switch off system notifications

I Switch off networking (unless necessary for the presentation)

I Do not point to the screen of the computer,but to the projected image

Clara Loh General advice for slides 16

Technical etiquette

I Bring the correct plugs to connect computer and projector

I Connect computer and projector ahead of time

I Know how to switch to fullscreen/presentation mode

I Switch off system notifications

I Switch off networking (unless necessary for the presentation)

I Do not point to the screen of the computer,but to the projected image

Clara Loh General advice for slides 16

Technical etiquette

I Bring the correct plugs to connect computer and projector

I Connect computer and projector ahead of time

I Know how to switch to fullscreen/presentation mode

I Switch off system notifications

I Switch off networking (unless necessary for the presentation)

I Do not point to the screen of the computer,but to the projected image

Clara Loh General advice for slides 16

TALK OVER!

blackboard preparation time < 60 minutesslides preparation time > 600 minutes

Try again?

Bonus game: General advice for talks

Theorem (Fundamental theorem of presentations)

Well-structured talks are easy to present and comprehend.

Level 0: Research

I Obtain interesting results

I Check correctness of results

Level 1: Raw concept

I Decide on the main goal of the presentation

I Sketch the main structure of the talk

Clara Loh Bonus game 18

Bonus game: General advice for talks

Theorem (Fundamental theorem of presentations)

Well-structured talks are easy to present and comprehend.

Level 0: Research

I Obtain interesting results

I Check correctness of results

Level 1: Raw concept

I Decide on the main goal of the presentation

I Sketch the main structure of the talk

Clara Loh Bonus game 18

Bonus game: General advice for talks

Theorem (Fundamental theorem of presentations)

Well-structured talks are easy to present and comprehend.

Level 0: Research

I Obtain interesting results

I Check correctness of results

Level 1: Raw concept

I Decide on the main goal of the presentation

I Sketch the main structure of the talk

Clara Loh Bonus game 18

General advice for talks, continued

Level 2: Detailed concept

I Be precise

I Be concise

I Put important stuff at the beginning (so it will not get lost in time)

I Attribute results correctly

I Explain what your contribution is

I Highlight important ideas/steps instead of technical details

I Try not to end with a subtle technical detail,but rather with an open problem or a conclusion

I Prepare for the likely case that the talk is longer than expectedand for the unlikely case that the talk is shorter than expected

Clara Loh Bonus game 19

General advice for talks, continued

Level 3: Presentation

I Memorize the first few sentences of the talk

I Be independent of your notes; look at the blackboard instead(This will reduce the risk of typos/gaps)

I Give the audience enough time to digest your arguments

I Keep eye-contact with the audience

I Try to get the audience involved

I Don’t be afraid of questions!

Clara Loh Bonus game 20